Octahedron - Area and Volume

Area and Volume

The surface area A and the volume V of a regular octahedron of edge length a are:

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

If an octahedron has been stretched so that it obeys the equation:

The formula for the surface area and volume expand to become:

Additionally the inertia tensor of the stretched octahedron is:

I = \begin{bmatrix} \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\ 0 & \frac{1}{10} m (x_m^2+z_m^2) & 0 \\ 0 & 0 & \frac{1}{10} m (x_m^2+y_m^2) \end{bmatrix}

These reduce to the equations for the regular octahedron when:

Read more about this topic:  Octahedron

Famous quotes containing the words area and/or volume:

    “During the Civil War the area became a refuge for service- dodging Texans, and gangs of bushwhackers, as they were called, hid in its fastnesses. Conscript details of the Confederate Army hunted the fugitives and occasional skirmishes resulted.”
    —Administration in the State of Texa, U.S. public relief program (1935-1943)

    “She carries a book but it is not
    the tome of the ancient wisdom,
    the pages, I imagine, are the blank pages
    of the unwritten volume of the new.”
    —Hilda Doolittle (1886–1961)